Abstract

We introduce two new types of fixed point theorems in the collection of multivalued and single-valued mappings in complete metric spaces.

1. Introduction

Let be a mapping on a complete (or compact) metric space . We do not assume richer structure such as convex metric spaces and Banach spaces. There are thousands of theorems which assure the existence of a fixed point of . We can categorize these theorems into the following four types.(T1)Leader type [1]: has a unique fixed point and converges to the fixed point for all . Such a mapping is called a Picard operator in [2].(T2)Unnamed type: has a unique fixed point and does not necessarily converge to the fixed point.(T3)Subrahmanyam type [3]: may have more than one fixed point and converges to a fixed point for all . Such a mapping is called a weakly Picard operator in [3, 4].(T4)Caristi type [5, 6]: may have more than one fixed point and does not necessarily converge to a fixed point.

We know that most of the theorems such as Banach’s [7], Ćirić’s [8], Kannan’s [9], Kirk’s [10], Matkowski’s [11], Meir and Keeler’s [12], and Suzuki’s [13, 14] belong to . Also, very recently, Suzuki [15] characterized . Subrahmanyam’s theorem [3] belongs to , and Caristi’s theorem [5, 6] and its generalizations [15–17] belong to . On the other hand, as far as the authors do know, there are no theorems belonging to ; see Kirk’s survey [18]. Also, recently many interesting fixed point theorems are proved in the framework of ordered metric spaces; see [18–35] and others.

In this paper, motivated by the above, we introduce two new types of fixed point theorems in the collection of multivalued and single-valued mappings and will prove them, which belong to .

Let be a metric space, and let denote the class of all nonempty, closed, and bounded subsets of . Let be a multivalued mapping on . A point is called a fixed point of if . Set .

A famous theorem on multivalued mappings is due to Nadler [36], which extended the Banach contraction principle to multivalued mappings. Many authors have studied the existence and uniqueness of strict fixed points for multivalued mappings in metric spaces; see, for example, [37–44] and references therein.

Let be the Hausdorff metric on induced by ; that is,
Denote and , where .

2. Main Results

The following is the first our main results.

Theorem 1. Let be a complete metric space and let be a mapping from into itself. Suppose that satisfies the following condition:
for all . Then(a) has at least one fixed point ;(b) converges to a fixed point, for all ;(c)if are two distinct fixed points of , then .

Proof. Let be arbitrary and choose a sequence such that . We have
Given
we have
Observe that is nonincreasing, with positive terms. So and . It follows that
Thus, it is verified that
Now for all we have
Suppose that . Since
. It means that
as . In other words, is a Cauchy sequence and so converges to .We claim that is a fixed point.Note that
On taking limit on both sides of (11), we have . Thus, .If there exist two distinct fixed points , then
Therefore, and we find the desired results.

In the following, two examples of such type of mappings, which satisfy (2), are given.

Example 2. Let and let be defined by
is a complete metric space. Let be defined by
and we have
and also
Therefore, satisfies all the conditions of Theorem 1. Also, has two distinct fixed points and .

Example 3. Let be endowed with Euclidean metric and let be defined by
Then we claim that satisfies all the conditions of Theorem 1.If and , we have
Thus,
Similar argument holds for the other conditions.

Remark 4. Note that in (2) the ratio
might be greater or less than 1 and has not introduced an upper bound. Note that if, for every , , then we have
It means that
and thus Theorem 1 is a special case of Banach contraction principle. Therefore, when is a complete metric space such that, for all , , Theorem 1 is valuable because (20) might be greater than 1. Example 2 shows this note precisely.

The following is the second in our main results.

Theorem 5. Let be a complete metric space and let be a multivalued mapping from into . Let satisfy the following:
for all . Then has a fixed point .

Proof. Let and . For each one can choose such that
For each we can choose such that
Specifically if
then
Therefore,
It can easily be seen that
Thus, it is easily verified that
Now for all we have
Suppose that . Since
. It means that
as . In other words, is a Cauchy sequence and so converges to . We claim that is a fixed point. Consider
On taking limit on both sides of (31) we have . It means that .

Remark 6. Note that Theorem 5 is a generalization of Theorem 1 because by taking and applying Theorem 5 for we obtain Theorem 1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.